/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 33 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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(0)
Obligation:
Conditional term rewrite system:
The TRS R consists of the following rules:

   e(0) -> true
   o(0) -> true

The conditional TRS C consists of the following conditional rules:

   e(s(x)) -> true <= o(x) -> true
   e(s(x)) -> false <= e(x) -> true
   o(s(x)) -> true <= e(x) -> true
   o(s(x)) -> false <= o(x) -> true


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(1) CTRSToQTRSProof (SOUND)
The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   e(s(x)) -> U1(o(x))
   U1(true) -> true
   e(s(x)) -> U2(e(x))
   U2(true) -> false
   o(s(x)) -> U3(e(x))
   U3(true) -> true
   o(s(x)) -> U4(o(x))
   U4(true) -> false
   e(0) -> true
   o(0) -> true

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:o_1 > 0 > s_1 > U3_1 > e_1 > U1_1 > true > U4_1 > false > U2_1

and weight map:

   true=5
   false=6
   0=1
   e_1=5
   s_1=3
   U1_1=4
   o_1=4
   U2_1=2
   U3_1=1
   U4_1=2

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   e(s(x)) -> U1(o(x))
   U1(true) -> true
   e(s(x)) -> U2(e(x))
   U2(true) -> false
   o(s(x)) -> U3(e(x))
   U3(true) -> true
   o(s(x)) -> U4(o(x))
   U4(true) -> false
   e(0) -> true
   o(0) -> true




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES